I am of a different vein of thought than the comments above - there are a couple of comments about the triangle inequality and other things. These are very good, and necessary for a full proof, but I think that none of these are necessary for a geometric intuition of the situation.
The magnitude of a vector u can be interpreted in a could of different ways: as the length of an arrow from one point to a point displaced by u away, or as the distance between the origin and the point u, for example. Let's think of the latter.
So where are the vectors x of unit length in the real plane? These are all vectors on the unit circle - alternatively, each point on the unit circle can be thought of as the tip of a vector of unit length. Similarly, the tips of the vectors of length 3 lie on a circle of radius 3 in the plane.
We have two vectors here, one of length 3 and one of length 5. And we are trying to maximize the length of them being added together. Well, let's draw two circles on a sheet of paper, concentric, and think of the smaller one having radius 3 and the larger having radius 5. Then we can think to ourselves: hmm, I want to get the largest possible result vector. How can we get this?
Well, let's choose a vector of length 3. How? Well, mark a point on the smaller circle. That's a vector of length 3. Now we want to know what vectors can result when we add a vector of length 5. What's that mean? That's like drawing a circle of radius 5 around our 'chosen' vector of length 3. And we want to see how far we can get from the origin.
Do this a couple more times for a couple more points - one will see that despite its fancy name, the 'chosen one' is very insignificant. Any initial point chosen will give the same maximal answer. In any case, the answer should quickly become apparent (I recommend using a point on say, the x-axis as your first point). If you are clever or imaginative, then you might even see how this generalizes in higher dimensions readily. But with n-spheres instead of circles.
And while some consider this geometric idea nearly complete in proof, an actual proof that it's maximal (in my opinion) would rely on the triangle inequality or CS.