As indicated by Serre's reference list, a basic source of ideas about formal group laws would be the papers by Michel Lazard in the period 1955-1965. Just do a quick search in www.numdam.org using his last name. One of his shorter papers is *Sur les groupes de Lie formels à un paramètre.* Bulletin de la Société Mathématique de France, 83 (1955), p. 251-274. But I'm not sure these papers will be explicit enough to answer your question completely. Lazard was especially interested in dealing with $p$-adic fields, while Dieudonne dealt further with fields of prime characteristic. Certainly Lazard did more than anyone else to establish the abstract foundations of formal group laws.

Another possibly more readable source would be Bourbaki's *Groupes et algebres de Lie*, Chapters II-III (especially Chapter II) together with the historical notes at the end of that volume. Here in particular the connections with work of Dynkin and others, along with the Baker-Campbell-Hausdorff formula, are emphasized.

ADDED: I think Emerton's brief answer and the added comments essentially answer the original question; but I wanted to point to the broader background sources as well. Serre's 1965 lecture notes *Lie Algebras and Lie Groups* (Benjamin) end up in Chapter V of Part II at *Lie Theory*, with the initial convention: "Unless otherwise specified, $k$ will denote a field complete with respect to a non-trivial absolute value." In some places it is required that $k$ be of characteristic 0; then it may be allowed to be just a $\mathbb{Q}$-algebra.
For some classical Lie theory, $k$ is assumed to be the real or complex field.
But much is also done with analytic groups over ultrametric fields, etc. So it's important to keep track of which kind of base ring or field you are working over.

Theorem 3 on page 5.28 is the theorem in question here. Serre gives a complete proof but relies heavily on previous material. By now he is making a distinction between the "formal" case and the "analytic" case; it's essential to be in the "formal" case when you want to work over an arbitrary field of characteristic 0. With these qualifications, I'd agree that for Theorem 3 a relatively intuitive proof is possible avoiding much of the surrounding "analytic" material.