The sense of “line” here is
line, n. V.26.a. Track, course, direction; route
Oxford English Dictionary.
I looked in Life, Letters and Journals of Sir Charles Lyell, Bart., but didn’t find the letter from Lyell to which Darwin is replying. However, we can guess at the contents. Presumably Lyell had noted the problem of explaining the worldwide distribution of certain genera and families of fossil plants and animals. Darwin had discussed this in On the Origin of Species:
Scarcely any palaeontological discovery is more striking than the fact, that the forms of life change almost simultaneously throughout the world. Thus our European Chalk formation can be recognised in many distant parts of the world, under the most different climates, where not a fragment of the mineral chalk itself can be found; namely, in North America, in equatorial South America, in Tierra del Fuego, at the Cape of Good Hope, and in the peninsula of India. For at these distant points, the organic remains in certain beds present an unmistakeable degree of resemblance to those of the Chalk. It is not that the same species are met with; for in some cases not one species is identically the same, but they belong to the same families, genera, and sections of genera, and sometimes are similarly characterised in such trifling points as mere superficial sculpture. Moreover other forms, which are not found in the Chalk of Europe, but which occur in the formations either above or below, are similarly absent at these distant points of the world. In the several successive palaeozoic formations of Russia, Western Europe and North America, a similar parallelism in the forms of life has been observed by several authors: so it is, according to Lyell, with the several European and North American tertiary deposits. Even if the few fossil species which are common to the Old and New Worlds be kept wholly out of view, the general parallelism in the successive forms of life, in the stages of the widely separated palaeozoic and tertiary periods, would still be manifest, and the several formations could be easily correlated.
Charles Darwin (1859). On the Origin of Species, pp. 322–323. London: John Murray.
This was a century before the development of the theory of plate tectonics, so that the simultaneous appearance of similar species in the fossil record of the Old World (Africa and Eurasia) and the New World (the Americas) was a puzzle for geologists. In his letter to Lyell, Darwin suggests the idea that species migrated from one continent to another by taking a line (route) though Siberia, across the Bering Strait to Alaska (or vice versa). He saw that there was a severe problem with this theory, namely the simultaneous appearance of groups only found in the tropics, which could not have survived the passage through Siberia. This is why he suggests that it might have happened “when the climate was much hotter”.
Lyell preferred the theory of continental drift:
If we go back still further—to the terrestrial plants and animals of the Eocene period, we find such a mixture of forms now having their nearest living allies in the most distant parts of the globe, that we cannot doubt that the distribution of land and sea bore scarcely any resemblance to that now established. Continents therefore, although permanent for whole geological epochs, shift their positions entirely in the course of ages.
Charles Lyell (1867). Principles of Geology, tenth edition, volume 1, p. 254. London: John Murray.
