Linguist George Starostin published an Oxford Research Encyclopedia entry on the Altaic Language hypothesis in April of 2016, that is a good quick introduction to the subject from one of its leading proponents. In the barest nutshell (from the beginning of this entry):
“Altaic” is a common term applied by linguists to a number of language families, spread across Central Asia and the Far East and sharing a large, most likely non-coincidental, number of structural and morphemic similarities.
At the onset of Altaic studies, these similarities were ascribed to the one-time existence of an ancestral language—“Proto-Altaic,” from which all these families are descended; circumstantial evidence and glottochronological calculations tentatively date this language to some time around the 6th–7th millennium bc, and suggest Southern Siberia or adjacent territories (hence the name “Altaic”) as the original homeland of its speakers.
However, since the mid-20th century the dominant view in historical linguistics has shifted to that of an “Altaic Sprachbund” (diffusion area), implying that the families in question have not sprung from a common source, but rather have acquired their similarities over a long period of mutual linguistic contact.
The bulk of “Altaic” has traditionally included such uncontroversial families as Turkic, Mongolic, and Manchu-Tungusic; additionally, Japanese (Japonic) and Korean are also frequently seen as potential members of the larger Altaic family (the entire five branches are sometimes referred to as “Macro-Altaic”).Basically, the winds of scholarship seem to be drifting from a "hard Altaic" position, in which the members languages emerged tree-like from a common proto-language, to a "soft Altaic" position that sees the similarities between members of the language family as possibly due to borrowings between geographically adjacent language families.