How does feedback make a system stable even in the presence of an inherently unstable element in that?
"When an element is unstable , it doesn't mean it is unstable for all inputs. For certain inputs the block can still produce stable outputs."

At first, an "element" (a part) cannot be unstable. It is better to use the term "active block" or "amplifier". Hence, an amplifier can be unstable if it has feedback, which produces a closed-loop pole (pair) with a positive real part. This is true for all inputs because it is the feedback loop which produces self-excitement of the system - not the input signal.

"During feedback, the input given to the the block gets modified such that output doesn't blow up."

An unstable system can be stabilized if the overall negative feedback overrides (dominates) the positive feedback which would cause instability (without negative feedback). In your case (example), the gain block has an inherent positive feedback (causing a pole at +0.5). This gain block ist stabilized using - as shown - 100% negative feedback causing a total closed-loop gain of "2 (and a pole at "-0.5")