You can think of it in these two steps:

1. λI stretches every vector (the whole space, really) by a factor λ.

2. Saying that the function is not injective means you lose information: when you apply it on some object and get a result, you can't trace back what was the original object, as there may be several. (There is no inverse function, then). In linear algebra, this only happens because there is some direction of space where all the vectors get collapsed to zero.

In short, T-λI collapses some line of vectors to zero.

So, when you took the effect of λI from T, you make it a lossy transformation in some direction. This means that _in that direction_ T had the effect of stretching all vectors by a factor of λ.

You gain some geometric understanding of T.

It is sort of intuitive, but the language may obscure it a little if you are not used to it.