San Marino is a legacy of the era when Italy was a patchwork of city states, some relatively big, some as tiny as San Marino.

During the 19th century a group of Italian statesmen and revolutionaries started a movement to unify all of the peninsula under a single state. Needless to say that the governments of most of these small states weren't happy with this idea, and would often persecute Italian nationalists. San Marino had a policy of giving political asylum to Italian nationalists, so much so that when Italy was eventually unified, the victorious nationalists allowed San Marino to keep its independence. They might have been less generous if it had been a larger state, but it was so small and insignificant, both economically and culturally, that they could treat San Marino's continued independence as a symbolic gesture (in a way that, for example, allowing Parma or the Two Sicilies to keep their independence wouldn't have been).

Since Italian unification San Marino has kept its independence simply because subsequent Italian governments have had little reason to try to annex it. Even Mussolini didn't do so, although he did sponsor and abet San Marino's fascists in taking control of the government (although even local fascists kept San Marino neutral during WW2). Mussolini's refusal to annex San Marino was sometimes questioned, but he did so because its independence was a relic of the era of Italian unification, an era in Italian history that the fascist government generally saw positively and attempted to link itself to (in another symbolic gesture, the grandson of the hero of Italian independence, ran as a fascist candidate in elections in the late 1920s).

So, tl:dr - San Marino was independent prior to unification because it was just one of many tiny independent statelets, and after unification because it was seen as a symbol of the unification era.