I'm curious but skeptical. I would have thought that if there is a good and straightforward justification, it would be in the wikipedia page, but it's not. It simply says:
Quadratic voting is a variant of cumulative voting in the class of cardinal voting. It differs from cumulative voting by altering "the cost" and "the vote" relation from linear to quadratic.
Apart from that there's no comparison to cumulative voting, which seems a lot simpler and more robust against concerns like collusion and sybil attacks. The only other mention of "linear" is in a rather opaque paragraph marked "citation needed".
If it's purely a way to allow people to buy votes without having rich people completely corner the market, I guess it makes some sense, but it still seems rather wrongheaded. If you're going to give everybody the same number of virtual tokens, there are no rich people to begin with. If you insist on using an actual currency, there are other possible mechanisms like a universal basic income.
Edit to add: Aha, I've found a (the?) paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2343956
Rational agents maximize their utility by setting marginal cost equal to marginal benefit. This means that if John values being able to incrementally move the outcome in his favor twice as much as Sue values being able to incrementally move the outcome in her favor, John will pay twice as much at the margin as Sue does. For example, John buys 16 votes while Sue buys 8 votes. The exact number of votes that John and Sue buy depends on their estimates of how likely they will be pivotal voters, as explained below, so if John buys 16 votes for $256 (162 ), this does not mean that he values the project at $256. But it does mean that he values the project twice as much as Sue, who buys 8 votes.
This emphasis on marginal cost seems to ignore the fact that John and Sue will be working with finite budgets -- the overall real cost of the vote is important too! Unless I misunderstand, the implication is that if the cost were linear, John would buy an unbounded number of votes (as the marginal cost never goes up) which seems absurd.
Particularly in the cases where quadratic voting is actually being used in government, when e.g. lawmakers are given 100 tokens to split among different possibilities, I don't think any of the above analysis applies at all, and it's not clear that quadratic voting has any advantages over cumulative voting.