This asymptotic was stated by Hardy and Ramanujan, but without proof. The first proof of this asymptotic was given by Wright in 1934 [1], by a rather complicated argument. A much simpler approach using the circle method was given by Vaughan in 2015 [2]. Vaughan also gives much more information, allowing for more terms in this asymptotic expansion (see his Theorem 1.5). The introduction of Vaughan's paper gives more information. Vaughan's proof has been generalised to give similar asymptotic formula for the partition function restricted to $k$th powers for any $k\geq3$ by Gafni [3].
[1] E. M. Wright, Asymptotic partition formulae III. Partitions into $k$th powersAsymptotic partition formulae III. Partitions into $k$-th powers, Acta Math. 63 (1934), 143-191143–191. Project Euclid (scanned pdf).
[2] R. C. Vaughan, Squares: Additive questions and partitionsSquares: Additive questions and partitions, International Journal of Number Theory 11 (2015), 1367-14091367–1409. https://www.worldscientific.com/doi/epdf/10.1142/S1793042115400096doi:10.1142/S1793042115400096.
[3] A. Gafni, Power partitionsPower partitions, Journal of Number Theory 163 (2016), 19-4219–42. doi:10.1016/j.jnt.2015.11.004, arXiv:https://arxiv.org/abs/1506.06124.
