Most Vanguard funds, even the Admiral ones, only require $3,000 USD minimum these days. The expense ratio is often 0.15% or less. Index funds are as low as 0.04%.

https://investor.vanguard.com/mutual-funds/share-classes

Why is the initial investment so high? Is there a reasoning behind it?

He's talking about a specific class of funds that are sold generally to institutions. Think buying funds in "wholesale"; they have low fees, high minimums. Those same funds can be bought "retail" though, with normal fees and typical minimums ($3k-$10k).

which funds require this?

He's talking about the "Admiral" class of funds which have lower fees but higher minimums. (And most are $50k min, not 500k). Most of these funds can be bought at lower minimums too, they just have higher fees.

I'm invested in Vanguard 500 Index Fund Admiral Shares, and the minimum investment is $3000, with fees of 0.04%. Those are pretty low minimums, and seem to me to be exceptionally low fees, and the average annual return is 13.94% over the last 10 years (before taxes).

I think OP is referring to the "Institutional" class of funds. They have much higher minimums because their target market is pension funds and large 401(k) plans that invest using pooled accounts.