We are up to Tractate Zevachim, Chapter 11, Mishnah 5. To understand this mishnah and the next one, a basic understanding of how the laws of tumah and taharah apply to vessels is required. The fundamental principle is that a vessel is susceptible to tumah on its own as long as it can function as a vessel.
The problem before us:
When a vessel becomes tamei, there is a Torah prohibition against bringing it back into the Azarah. On the other hand, we have already learned that the removal of blood, absorbed flavors, or fat from vessels - which is essentially the chatas offering in all its respects - must be done inside the Azarah. What, then, is the halachah when a garment or vessel that had chatas blood on it (or absorbed chatas flavor and fat) was taken outside the Azarah, whether by mistake or even intentionally, and while outside it contracted tumah, and now it must be brought back inside for washing or scouring?
The solution - breaking the vessel:
The answer is that one must break the vessel. The definition of breaking a vessel is that it can no longer fulfill its original intended purpose, and therefore it loses its status as a vessel, and with that, it loses its tumah. For example: a shirt that had chatas blood fall on it and was taken outside the Azarah, which must be brought back inside for washing, and while outside it touched a dead mouse and contracted tumah - one tears it along the majority of its length, and thereby it becomes broken. It can no longer function as a shirt, and therefore it loses its status of tumah.
However, a problem arises here: The verse requires washing the garment, which is a vessel, in the Azarah. The answer is that the tearing is done across its majority and not completely. In this state, the vessel certainly cannot function in its original capacity, since its majority is broken, but it can certainly be designated for another purpose - shading from the sun, a baby blanket, or a patch for a torn coat. The fabric can be repurposed, and therefore it is susceptible to contracting tumah anew and it is a vessel or a potential vessel, similar to what we saw a few mishnayos earlier regarding animal hide that has not yet been decided if it will be used as is. And if it is indeed designated as a new vessel - it does not matter, because the old tumah has already departed.
The text of the mishnah:
"Beged sheyatza chutz laklaim" - A garment that had chatas blood on it, and therefore it must be washed inside the Azarah, and it was taken outside the curtains. "Klaim" - the curtains that surrounded the courtyard of the Mishkan, which correspond to the walls of the Azarah in the Beis Hamikdash, an expression we have seen several times. And its halachah is: "Nichnas umechabso" - He brings it back inside and washes it in the Azarah as required by the verse. (Regarding the grammar, one could analyze why the mishnah used the word "nichnas" rather than "machnis"; nevertheless, the intent is that it must be brought back inside.)
And if while outside the partitions, beyond the walls of the Azarah or the curtains of the Mishkan, it contracted tumah - it may not be brought in as is, due to the prohibition of bringing a tamei vessel into the Azarah. Therefore, he tears its majority, and once its majority is torn it loses its original status as a vessel and becomes tahor; and since it can still be used for another purpose, it retains the legal title of a garment, and therefore he brings it back into the Azarah and washes it as is.
An earthenware vessel:
We have already seen that an earthenware vessel that had leftover kodashim or their absorbed flavors in it - it must be broken, once it becomes nosar after being left overnight, past its allotted time. If that earthenware vessel was taken outside the partitions, beyond the curtains and the walls of the Azarah - it must be brought back inside and broken inside the Azarah. And if while outside it contracted tumah, what should he do?
Why is immersion in a mikvah not a solution?
Seemingly, the first response to a situation where the garment is ritually impure and cannot be brought into the Azarah is to immerse it in a mikvah, so that it will become pure and can enter. However, immersing the garment in a mikvah actually begins the laundering process, since some of the blood dissolves in the water, and it is forbidden to perform any laundering outside the Azarah - a Torah prohibition, since the laundering must be done inside the Azarah as we discussed. Therefore, a mikvah is not an option. And for an earthenware vessel, a mikvah is certainly not a solution, since a mikvah does not purify earthenware vessels at all, and only breaking it purifies it.
One cannot break it completely either, because then it would lose its status as a vessel, and it would not be the vessel that is returned and broken in the Azarah. Rather, a small hole is punctured in it. The potter who creates an earthenware jug does not know exactly what will be placed inside it, but certainly wants it to be able to hold liquids - like a pickle jar that holds its water - and earthenware vessels are not manufactured with a hole in their bottom. Thus, their function is limited and defined.
Therefore, an earthenware jug that has remnants of charcoal left in it, whose law is to be brought and broken in the Azarah after its time, and it became ritually impure: if a small hole is punctured in its bottom, it ceases to be usable for its original purpose for which it was created, and therefore it loses its tumah; yet it can be repurposed to hold things larger than the hole - dried figs, pencil erasers, gumballs, and the like - and because of this, it still retains the status of a vessel.
This is the law of the Mishnah: an earthenware vessel that went outside the enclosures - one brings it back and breaks it within the boundary. If it became ritually impure while outside, such as if a dead mouse fell into it - one punctures a small hole in its bottom, and once punctured it no longer functions in its original manner, losing its definition as a vessel and with it its tumah, and then it is brought back to the Azarah and broken there to fulfill the obligation.
In summary: In this Mishnah, we learned the principle that a vessel can contract tumah as long as it is fit for its purpose, and that breaking it removes its status as a vessel and along with it, its tumah. From this comes the course of action for a garment that went outside the curtains and became ritually impure - tearing the majority of it, which leaves it fit for another purpose and therefore still a garment requiring laundering inside; and for an earthenware vessel that went outside and became ritually impure - puncturing a small hole, which removes its original functionality and leaves it fit for another use, and thereby it can be brought back and broken in the Azarah. We also established that immersing in a mikvah is not a solution: for a garment because it begins the laundering process outside the Azarah, and for an earthenware vessel because a mikvah does not purify it at all.
In the next part, we will focus on the following Mishnah, which continues with these principles of the tumah of vessels and repurposing them.