This is a symbolic expression, by which Willy means that his world is being destroyed. He is financially hard-up, he is on the verge of losing his long-term job as a salesman, and he is nervous and exhausted. He not only feels himself a failure, but his sons too seem to be going the same way. His only refuge is to think back to the past, when his sons were still young and the future seemed more promising (although even then he was struggling to make money).
Willy makes the remark, 'The woods are burning' to his younger son Happy, who was trying to cheer him up by promising him that 'I'm gonna retire you for life'. Willy is openly scornful of this, as Happy isn't making much money either and what he does make goes to paying rent on his apartment and on women. His other son Biff doesn't even have a regular job. His sons' lack of success makes Willy feel even worse. Willy feels that his life has turned out wrong and that it's all going to end in disaster - as indeed it does, when he ends up committing suicide.
Other people generally don't understand Willy as he tends to live in his own world of frustrated longings, shattered dreams, and constant harking back to the past. A remark such as 'the woods are burning' doesn't make much sense on the face of it, but it is an expressionist touch, a sign of the incessant despair and turmoil inside Willy. Trees and woods are important symbols of the play as a whole. Willy wistfully remembers the old days when there used to be trees in the neighbourhood; now the area is all built up and he feels 'boxed' in. Trees represent the natural world which Willy feels is being destroyed by modern, materialistic, competitive urban society - a society where he himself cannot cope. He is being destroyed along with the old, natural world. He and his sons all express a liking for 'the great outdoors', and we are constantly reminded of the fact that he is far more skilled as a carpenter than as a salesman. In short, he is in the wrong job.
When Willy says to Charley, his neighbour, 'I got nothing to give him, Charley. I'm clean, I'm clean' - he is talking about his son Biff. Biff is a deeply troubled man who has a strained relationship with Willy. He blames Willy a good deal for having foisted his own unrealistic dreams and expectations of great material success upon his sons. Yet, in the past, Biff used to idolize Willy. The changed relationship between them is another source of torment for Willy. Biff also despises Willy for having had a secret love affair.
Willy feels terrible that Biff has turned out to be so unsuccessful in life, and bemoans the fact that he himself isn't rich enough to help him out financially - 'I got nothing to give him.' When he says, 'I'm clean', this means that he has been cleaned out, he has no more money; he is empty, stripped down to the bare bones financially. He might also mean that he is also spiritually and emotionally drained by this time; he is unable to offer Biff any kind of genuine moral encouragement and support.
"The woods are burning" because "the woods" represent the life path Willy didn't choose. Literally the timberland Ben offered him to manage. Looking back, he is confident in his older age that position was his golden ticket to success. One of his only hopes left in life is that he and his sons can find the figurative timberland in New York that will secure the success that passed him by. As his sons age and seem to be going in circles, he sees the opportunity for success forsaking them as well, and every day the woods burn more.
The "I'm Clean!" remark is a reference to his regret for being unfaithful to his wife. The suspicion that Biff hates him for being caught with another woman won't leave his mind. Believing that this caused Biff to go astray is too much for him to bear, but the thought haunts him. He makes this statement as he is playing a poker game in which his mind is torn between past and present. This statement is part of the struggle to communicate with Charley and shadows of the past simultaneously. He gets taken off to the past and answers Ben's memory, and has to try and fit these words into his talk with Charley.