察する means "to pick up on the hidden nature of something (often, a situation or feelings) without it being made explicit for you" (hence "infer", "surmise", "guess" -- although I think we're getting pretty far from the original meaning at that point).
You can 察する that it is not appropriate to speak, for instance (i.e. 空気を読む). You can also use it like 少しは察しろ ("use some tact", "realize how I feel", etc).
In お察しいたします, the speaker says he understands something intuitively -- the listener's feelings. So this is a statement of sympathy.