Answering from the far future! I present some literature on this topic:

1) K. Dickman in his original paper of 1930 gave an heuristic argument that can be found in pages 382-383 of *The art of computer programming, volume 2* (third edition) by Knuth.

2) V. Ramaswami made the argument rigorous in his 1949 paper *[On the number of positive integers less than $x$ and free of prime divisors greater than $x^c$*][1].

See also the review [MR0031958][2] for links to "weaker" papers (according to the reviewer) of Chowla and Vijayaraghavan and of Buhštab published around the same time, plus further work of Ramaswami on the question.

3) The second part of the AMS memoir *Numbers with small prime factors, and the least $k$th power non-residue* by K.K. Norton (1971) surveys Dickman's function.

4) As already mentioned by Dimitris Koukoulopoulos, the paper [*Integers without large prime factors*][3] by Hildebrand and Tenenbaum (1993) is a survey which actualizes that of Norton.

5) *[Integers without large prime factors: from Ramanujan to de Brujin][4]* by P. Moree (2014) is a small recent survey highlighting the contributions of de Brujin to the question.

6) The very recent paper [*Exact asymptotics of positive solutions to Dickman equation*][5] by Diblík and Medina (2018) improves the well-known asymptotic formulas for Dickman's function.

7) Also very recent are tight estimates for effective counts of smooth numbers due to Lichtman and Pomerance (2018), published in [*Explicit estimates for the distribution of numbers free of large prime factors*][6].

8) A modern textbook reference for Dickman's function theorem is Theorem 7.2 of *Multiplicative number theory I: classical theory* (2006) by Montgomery and Vaughan.


  [1]: http://www.ams.org/journals/bull/1949-55-12/S0002-9904-1949-09337-0
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=31958
  [3]: http://jtnb.cedram.org/item?id=JTNB_1993__5_2_411_0
  [4]: http://math.colgate.edu/~integers/vol14a.html
  [5]: http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=14695
  [6]: https://www.sciencedirect.com/science/article/pii/S0022314X17303402