Think of the affine combination as a linear combination of position vectors, which we want to specify a point. Now, a position vector has two parts; a base vector to whatever point we're calling zero, and a displacement from that. Suppose you have coordinates based on the street grid numbers in a city - the vector says to go to the "zero point" downtown, and then move away from that a specified amount in each direction.
What happens when we apply an affine combination to these? Every vector in the combination has the same base vector, so we add a total of $1$ times that. The displacement vectors vary, and we get a new displacement vector from that. Overall, it's a position vector in the same form.
If, instead, we applied a linear combination without requiring the sum of coefficients to be $1$, we would multiply the base vector by that sum, whatever it is. Our new position vector would go to somewhere completely new as its base, then displace from there.
So what that sum condition means? It means that we can take an affine combination of position vectors for various points and get the position vector for a new point, in a way that doesn't depend on exactly which coordinate system we were using. Translating our coordinate system (choosing a new base point) translates that affine combination in exactly the same way, so that it still represents the same point.