"For that reason, let a prince have the credit of conquering and holding
his state, the means will always be considered honest, and he will be
praised by everybody because the vulgar are always taken by what a thing
seems to be and by what comes of it; and in the world there are only the
vulgar, for the few find a place there only when the many have no ground
to rest on."
As you have probably understood, "vulgar" here has its original Latin meaning, vulgus, the mass of ordinary, unrefined people.
the vulgar are always taken by what a thing seems to be and by what comes of it -- The masses always believe appearances and results, and don't think about what may be happening underneath, or about how many died during the successful conquest.
in the world there are only the vulgar -- in the "world" that the Prince must take into account, everybody belongs to the masses. That is, the Prince need not consider anything except the reaction of the masses.
for the few find a place there only when the many have no ground to rest on -- For=because. The "few" are those few who do not belong to the masses. In the real, political world, there is very little space for the "few", and only where the "many" (masses) either (a) leave them a bit of room, or (b) find themselves in deep trouble and turmoil. I think (b) is more important: the "few" who shake the ignorance of the masses can only do so when the times are troubled enough that the masses are upset. But of course the Prince has a responsibility not to allow such turmoil while he rules.