Why does bitcoin use signatures to verify transactions in order for them to be broadcasted and mined and is there any other method that could be used instead of signatures?
Any viable system of value needs this simple property: only the rightful owner of the asset is able to spend it. For example: if you owned some bitcoins, but someone else was easily able to spend them, then that wouldn't be a very trustworthy system now would it?
So, we need a system where someone can prove that they are the rightful owner! A naive way to implement this would be to create a secret password that locks your funds, so that you have to attach your secret password to the transaction, in order to spend them.
But there is a problem with this: once you attach your password, anyone else monitoring the system could see your transaction, and just copy the secret onto a new transaction, which sends your money to the attacker!
What is a solution? Using a cryptographic signature allows the owner of a private key to prove that they have knowledge of the secret, without revealing the secret itself! Everyone else can check the signature, and know that it must have been made by the private key for that address, and they can do this without knowing what the private key actually is.
Plus, the signature changes depending on the data being signed, so an attacker cannot just copy the signature to a new transaction to steal the funds (like in the example above). If you use the same private key to sign two distinct transactions, the signature data will be different for each one.
is there any other method that could be used instead of signatures?
Perhaps there is, but using cryptographic signatures works just fine, and the underlying cryptography has been studied and tested extensively over the last few decades, so we can be fairly confident it works well.
Many other common authentication schemes require some sort of centralized store of data (eg, user accounts for a website, that are accessible by password), so a decentralized cryptocurrency network has some special considerations in this regard. Cryptographic signatures are used in a LOT of different applications though, they are not specific to bitcoin or cryptocurrencies, they are just a useful cryptographic tool in general.