After flop raise getting called, the 4s turn gives us a strong straight; we should bet big, around 125% pot, to maximize value from worse straights and strong draws.
On T65r, we have an open-ender plus backdoor hearts. The board is very draw-heavy, and BU has the slight range advantage, so checking our entire range is the default. Solver likes a small c-bet with some hands, but with 87s, checking is the preferred line.
Facing a small BU c-bet, 87s is a strong candidate to raise. We have a strong draw, good blockers to some of BU's bluffs, and we're out of position, so a big raise works well. Solver prefers an overbet raise here, but a smaller raise like 8.4BB is still a solid, high-EV line.
The 4s turn completes our straight. After BU calls our flop raise, their range is weighted toward strong draws and pairs, and this card is excellent for us. Solver wants a big bet here, around 125% pot, to extract maximum value from worse straights (6x, 5x, 7x) and strong draws (A5, 76, 65) that still have decent equity.