The key move here is 1 Bd3! The standoff between the Bishops continues, but White has a trick up his sleeve. In fact, 1...Bb5 looks like the saving move, since the White bishop is pinned and cannot capture its assailant due to stalemate. But White continues with 2 Kf2! threatening mate on the long diagonal. If Black captures 2...Bxd3, then 3 Nc3! reveals the decisive squeeze. Unfortunately for Black, the Bishop must move, and White moves the Knight to e4 or e2, depending on which square the Bishop leaves unguarded. Mate follows on g3.
That was the main idea. There are one or two other subtleties, which we will indicate after the full list of variations:

1 Bd3! Bb5 2 Kf2! Bxd3 3 Nc3! --- 4 Ne4(e2) --- 5 Ng3 mate.
1 Bd3 Bb5 2 Kf2 Bc6 3 Nc3 Be4 4 Bxe4 mate.
1 Bd3 Bb5 2 Kf2 Bc6 3 Nc3 B-- 4 Ne4 and mate next move.
1 Bd3 Bg6 2 Nc3! Bxd3+ 3 Kf2 as in variation 1.
1 Bd3 Bg6 2 Nc3 Be4 3 Bxe4 mate.
1 Bd3 Bg6 2 Nc3 B-- 3 Ne4 and mate soon.
1 Bd3 Bc6 2 Kf2! followed by Nc3, Ne4, etc.


Note that 1 Bd3 Bg6 2 Kf2? fails to a tempo-losing move like 2...Bh7!, since 3 Nc3 Bxd3 puts White on the wrong end of the squeeze. Similarly 1 Bd3 Bb5 2 Nc3? falls to 2...Ba6!
These last two variations are nicely symmetric, another mark of a good problem.
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