Why cant opposition be ignored if the king cannot capture the other king?
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Lets say I have the black king on d6, a black pawn protecting it on e7, a white pawn on b4, and the white king on c4. Why cant white move the king to c5 next to the black king, seeing as the white pawn makes it to where if the black king captures the white king, it would be under check from the white pawn, and therefore be an illegal move, making it impossible to take the white king, therefore not placing it under check, and vice versa for the black king and pawn.
This is common question.
The simple answer is: the rules say that you can't put your K in check, regardless of the circumstances, regardless of whether your opponent could actually follow through with capturing your K.
A more explanatory answer is: you want to be allowed to put your K in check, but the reason you say that this is ok is that your opponent can't capture your K, because they're not allowed to put their K in check. The problem here is that, if we're suspending the rule about Ks being in check, that would apply to both players, not just for you.
So, pretend that Ks are allowed to be in check and the first player to capture the opponent's K wins the game. Follow the sequence of moves that you propose, and you'll see that your K would be captured first, ending the game. It wouldn't matter that your opponent's K was in check in this final position, because the game would have already ended before you could capture their K.
"The Opposition" is an endgame technique used in a variety of scenarios to prevent the enemy King from being able to intrude or encroach into your position for the purpose of either capturing your pieces (typically pawns, but sometimes other pieces may be the object), or preventing you from making progress with them as for example when you are trying to get a passed pawn to the queening square. Opposition therefore has nothing to do with Kings capturing Kings (which, of course, is not possible in chess).
The following video explains the Opposition in its most basic scenario - a K+P vs K endgame - where one side is attempting to Queen his lone pawn. First understand how Opposition is used to prevent the enemy King from making progress to prevent the queening of your pawn and you will begin to get the idea behind Opposition. Opposition can be used similarly in situations where you have, for example, multiple pawns on the board and you use Opposition to prevent the enemy King from encroaching into your position in an attempt to capture or stymie the progress of your pawns.
This video by John Bartholomew demonstrates the most basic Opposition scenario and technique......(the procedure starts at minute 4:00)...
https://www.youtube.com/watch?v=aLyRWZPXUzI
The following book (Chapters 1, 13, 14 & 15) is an excellent resource for learning about the Opposition and how to employ it in various scenarios...
Endgame Workshop by Bruce Pandolfini...
https://www.amazon.com/Endgame-Workshop-Principles-Practical-Pandolfini-ebook/dp/B0033AGA6C/ref=sr_1_1?crid=UVN7KHWBLG8O&dchild=1&keywords=endgame+workshop&qid=1620792833&s=books&sprefix=endgame+workshop%2Caps%2C208&sr=1-1
https://www.chess.com/blog/RussBell/scribd-com-for-online-chess-book-reading
Also, when studying the Opposition it is important to understand (i.e. perceive) when it is necessary for you to advantageously "take" the Opposition or "relinquish" the Opposition. That is, moving the King into, versus out of, a state of Opposition. This implies that you must also understand when to allow your opponent to be able to "take" (move into) the Opposition and when to force your opponent to ''relinquish" (move out of) the Opposition. So when studying and playing positions involving Opposition, always ask yourself who, with the move, (i.e., you or your Opponent) must either "take" or "relinquish" the Opposition. Thus there are opportune situations where you may want to take the Opposition and/or force your opponent to relinquish the Opposition (or vice-versa). You must understand how to handle these situations. After spending some time studying various Opposition scenarios, these questions and their answers, will become second nature to you.
Finally, you will need to know how to use Opposition in various geometries, for example, distant opposition, rectangular opposition and diagonal opposition. They all require an odd number of squares separating and along a line connecting the opposing Kings, and the Kings must also stand on the same color squares in order be said to be "in Opposition".
Using Opposition in a R+K vs K endgame...
https://www.youtube.com/watch?v=Jd0meolN7w
more about Opposition...
https://www.youtube.com/results?search_query=the+opposition+chess