Bcrypt is backed by Blowfish, designed by Bruce Schneier. Go look it/him up. It's secure.
MD5/SHA1/etc are not weak because they are cryptographically weak (though some are), it is weak because they are fast. SHA3, when it is picked, will still be a very bad choice because it too will be fast.
So, why Bcrypt? Well, it uses Blowfish. Blowfish has a very slow key scheduling algorithm which basically involves a lot of hashing to get the round subkeys. Bcrypt makes this even slower. So what? Well, with Bcrypt you could set it up to take .3 seconds to verify a password. Try bruteforcing on that.