Lawrence lets us into pretty much everyone's minds: Clifford, Connie, and Mellors, of course; but also Mrs. Bolton, Hilda, Connie's father, and even the Venetian gondolier Giovanni: "he got a mate to help him, for it was a long way; and after all, they were two ladies. Two ladies, two mackerels! Good arithmetic! Beautiful ladies, too! He was justly proud of them" (17.59).
One of the narrator's main tricks is swooping in and out of characters' minds with a technique called free indirect discourse. This is a little trick that's used a lot in 19th- (and 20th-) century novels: it combines elements of the narrator's voice with elements of the character's voice without using quotation marks or words like "he said" or "she thought," called tags.
Check it out:
But Clifford only smiled a little uneasily. Everything was ridiculous, quite true. But when it came too close and oneself became ridiculous, too. (1.31)
The sentence "But Clifford only smiled a little uneasily" is pure narrator. The narrator is objectively describing—or, well, kind of objectively—what Clifford is doing. And now for the next sentence: "Everything was ridiculous, quite true." There's no tag to tell you that this is what Clifford is thinking. You have to figure it out yourself from words like "quite true," which isn't what the narrator is saying, or even would say. It's a characteristic of Clifford's own voice.
Or this one:
Connie returned to her room, threw her pyjamas on the tossed bed, put on a thin tennis-dress and over that a woollen day-dress, put on rubber tennis-shoes, and then a light coat … For the rest, the only danger was that someone should go into her room during the night. But that was most unlikely: not one chance in a hundred. (13.266)
Again, the first sentence is just description. The narrator is telling us in exhaustive detail what Connie is doing and what she's putting on. But then the passage moves from Connie's exterior to her interior, her thoughts. When that happens, the narrator backs off a little. He's still controlling the whole thing, but he's giving the impression of letting us have total access to Connie's interior thoughts without directing them by saying something like "She thought that it was unlikely anyone would come to her room."
This technique of giving lots of people voice is typical of modernist works. It helps undermine the objectivity of the narrator and generally lends to the confusing, unsatisfying, but also rewarding experience of reading the novel.