The reason for Be2 is simple. You do not want to play f3 too soon. Do it when provoked by ...Re8. Otherwise, there could be Ne4 and Bh4 tricks looming (or Qh4 if the Bishop moves). Not saying concrete threat here, but the concept makes an early f3 without the Bishops developed risky.
Therefore, where does White want his GOOD Bishop, assuming f3 will get played eventually? Well, with f3, your good bishop is the dark squared one. It wants to go to e3.
Without f3 played and without Be2 played, Be3 gets harassed by ...Ng4.
Therefore, White plays Be2 to stop Ng4, then Be3, Qd2, and O-O-O, pausing to play f3 when provoked and not until then.
As mentioned earlier though, Bf4 is the strongest. The e3-response for the Bishop is on the basis that you are playing the slower strategy with Be2.
I have an ongoing correspondence game at ICCF with 6.Bf4.
We are 25 moves in and the position is fairly equal, maybe a tiny edge for me (white).
OK