Just write $SU(2)$ in some local coordinates (some of the standard systems are the double-polar system (a.k.a Hyperspherical coordinates) or the single angle coordinates (a.k.a Hopf coordinates) and then one sees that the special unitary condition forces the $4$ parameters you will need to satisfy the $3$-sphere equation. Hence one gets an explicit local map to the $3$-sphere by mapping that matrix to the ordered tuple of $4$ numbers. Smoothness is assured since the functions are all polynomials.

These coordinate systems make what MTS said below explicit.

After all one is very likely to want to do some geometry on $SU(2)$ now that one knows it is $S^3$ and these coordinate systems are naturally adapted to do them. Like computing vielbiens on $SU(2)$ in these systems look very natural and make the symmetries of the spherical structure underneath very clear.

A related extra stuff:

One can look up a very nice analysis of this in the first chapter of Gregory Naber's book "Geometry, Topology and Gauge Fields" Volume 1.

In that section he will do what gets called the Heegard Decomposition of $S^3$ using very simple high-school maths!

Basically Naber will rationalize why these coordinate systems are in some sense natural.

You can also look up these analysis in these two write-ups I had done during my undergrad , this and this one.

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Just write $SU(2)$ in some local coordinates (some of the standard systems are the double-polar system (a.k.a Hyperspherical coordinates) or the single angle coordinates (a.k.a Hopf coordinates) and then one sees that the special unitary condition forces the $4$ parameters you will need to satisfy the $3$-sphere equation. Hence one gets an explicit local map to the $3$-sphere by mapping that matrix to the ordered tuple of $4$ numbers.

One can look up a very nice analysis of this in the first chapter of Gregory Naber's book "Geometry, Topology and Gauge Fields" Volume 1.

In that section he will do what gets called the Heegard Decomposition of $S^3$ using very simple high-school maths!

You can also look up these analysis in these two write-ups I had done during my undergrad , this and this one.