23:16-22 How are we to understand what Jesus says here?

First of all let us see what a vow is in this context. It is a solemn undertaking to do something for God in return for some specific benefit from Him (cp Gen 28:20-22).

The scribes and Pharisees in Mt 23:16-22 were evading paying their vows by dishonestly applying a false set of conditions to swear by which ensured that it was never binding upon them to pay (cp Mt 15:1-9; Mk 7:5-13).

Jesus rebuked the scribes and the Pharisees for being so devious and told them that regardless of what they swore by, the vows had to be paid (cp Deut 23:21-23; Psa 66:13-16; Ec 5:4-5 with Judges 11:30-40).