Polonius is simply responding to Claudius' assessment of his character: a man 'faithful and honourable'. 'Fain' means 'happily', so Polonius is simply saying: 'I would be very happy to be proved to be as good as you say I am'.
The comparison to Jepthah is more complex: Jepthah is an Old Testament Judge (see the book of Judges, below) who promised God that, in return for victory in battle, he would sacrifice the first creature that came to him when he returned to his native land. Of course, it was Jepthah's daughter who greeted him, but the Old Testament is unclear about precisely whether or not Jepthah did go through with the sacrifice. The mythological figure of Jepthah does sacrifice his daughter, but commentators disagree about the biblical one.
Might we see Polonius as sacrificing his daughter? He certainly isn't cautious about planting her where Hamlet might discover her, so that he and Claudius can spy on Hamlet (who is, so they think, mad). He also reads Hamlet's love letter to her out without any regard for her privacy or feelings.
Is it Polonius' fault that Ophelia goes mad and eventually commits suicide? It depends entirely how you interpret the character: Shakespeare is very ambiguous about exactly what sends Ophelia insane. But the reference to Jepthah opens up the possibility that Polonius, the wretched, rash, intruding fool, might - where Ophelia is concerned, at least - be a rather more sinister character.
Jephthah was one of the judges in the biblical book of Judges. He promised God that if he had victory in battle, he would sacrifice to God the first creature who came running to him when he returned home. Jephthah expected that creature to be his dog, but instead his daughter came running out to greet him. God did not ask for this sacrifice, but the daughter resolves to help her father keep his word by willing going to her death.
Likewise, Polonius is Ophelia's father. Some might say that he sacrifices her to further his own political ambitions. Ophelia's suicide can be compared to the death of Jephthah's daughter as well.
Polonius says "I would fain prove so" in response to Claudius's answer to what he thinks of Polonius. Claudius's answer is, "a man faithful and honorable." "Fain" can mean "gladly," so Polonius is saying, "I would gladly prove so."