You have identified one of the themes implicit in this poem. What is interesting though is that loneliness is normally portrayed as a negative emotion, but here, Frost definitely seems to portray it as something that the persona of this poem craves and wants. The first stanza tells us that he is alone by some woods, with the owner of those woods far away:
He will not see me stopping here
To watch his woods fill up with snow.
The speaker seems to be entranced by the beauty of the falling snow and the woods, which is why he is stopping on his journey by himself at night.
The key stanza is of course the last stanza, which makes evident the speaker's attraction of the woods and his desire to stay, but also the recognition that he must move on:
The woods are lovely, dark, and deep,
But I have promises to keep,
And miles to go before I sleep,
And miles to go before I sleep.
This, to me, makes the symbolic meaning evident - Frost is writing about the desire to die but at the same time he recognises that it is not time yet as he has a lot of responsibilities that he needs to fulfil. However, he is free to enjoy the emotion of loneliness at times like this where he takes "time out" from the responsibilities of life to reflect on the beauty of nature.
In this poem, isolation is something that the writer craves and desires, and attempts to chase and capture for a few brief moments for himself. If you look at the last lines, he ponders how he cannot stay, because he has "promises to keep" and "miles to go" before he can rest and get some sleep. He feels pressured to keep going; he has an errand; he has to meet someone or some people and fulfill his duties. So, in a brief moment of desperate desire for quiet and peace, he stops, in the middle of nowhere, just to be alone.
Whereas isolation is a word that typically has negative connotations, the writer of the poem seems to crave it. He mentions how even the horse finds it odd to be stopping "without a farmhouse near," and grows impatient to get going again. The "the sweep of easy wind and downy flake" are the only sounds there, and Frost calls that "lovely." He even knows the man's fields, but this is his chance to get a few minutes alone. Perhaps he feels too connected; in fact, he makes sure to mention that the man whose fields these are will not see him, because he owns a house in town. The fact that Frost mentions that alludes to the fact that he purposefully stopped somewhere where he could be completely and totally isolated and unobserved. He revels in that for a moment before the pull of obligations moves him on.
To Frost in this poem, isolation is a rare beauty that he must cut out of his busy, obligated schedule in small moments. I hope that helped; good luck!