To provide support for Jesus's claims, Jesus brings up witnesses in a style similar to the one used in the Mosaic Law. First up is John. Back in John 1, John talked about how he saw the Holy Spirit came upon Jesus, so Jesus counted John as a witness. As stated by Jesus (and also John), John was "the burning and shining lamp" (5:35) to convey the idea that John was providing guidance (and verification, too). In fact, some of the Jewish leadership had experience with John (5:35), suggesting that John's witness is something familiar to some of them.
It's worth noting that in 5:34 that Jesus "do[es] not receive testimony from man". Essentially, Jesus was saying that he is not reliant on man's testimony to the point that he has to seek witnesses. Note that the Greek for "receive" (Strong 2983) can also convey the meaning of "strive to obtain", and some translations (ICB, NOG, NLT) support the interpretation. However, Jesus brought up John as witness so he can provide evidence for the Jews. As we'll see in the next few verses, Jesus has a few other witnesses that he can show as evidence to support the legitimacy of Jesus's claims.