Isn't it curious that when Vernon Dursley was talking about magic all the time, or when the owls came to their house, that the neighbors never came knocking to see what was going on? Or perhaps that it only happened to the Dursleys in their neighborhood. Also, when Harry was escaping in the Duel over Privet Drive in the beginning of Deathly Hallows, no one heard them talking or the battle happening when they descended far enough for the muggles to see them in the sky. Also, when wizards were going to London while the muggles were there, no one noticed how strange they behaved or how they disappeared into the Ministry after flushing themselves down a toilet. Or when Aunt Marge got blown up, and the neighbors never noticed.
When owls were flying about on a much larger scale at the beginning of the first book (after Voldemort's alleged death), Muggles interpreted it as something merely unusual, yet not caused by magic:
“And finally, bird-watchers everywhere have reported that the nation’s owls have been behaving very unusually today. Although owls normally hunt at night and are hardly ever seen in daylight, there have been hundreds of sightings of these birds flying in every direction since sunrise. Experts are unable to explain why the owls have suddenly changed their sleeping pattern.” The newscaster allowed himself a grin. “Most mysterious. And now, over to Jim McGuffin with the weather. Going to be any more showers of owls tonight, Jim?”
“Well, Ted,” said the weatherman, “I don’t know about that, but it’s not only the owls that have been acting oddly today. Viewers as far apart as Kent, Yorkshire, and Dundee have been phoning in to tell me that instead of the rain I promised yesterday, they’ve had a downpour of shooting stars! Perhaps people have been celebrating Bonfire Night early — it’s not until next week, folks! But I can promise a wet night tonight.”
If Muggles don't come up with a good explanation for the phenomenon, then there's a part of the Ministry dedicated to creating good explanations for high-impact magical phenomenon, as explained by the Fantastic Beasts book:
The Office of Misinformation will become involved in only the very worst magical-Muggle collisions. Some magical catastrophes or accidents are simply too glaringly obvious to be explained away by Muggles without the help of an outside authority. The Office of Misinformation will in such a case liaise directly with the Muggle prime minister to seek a plausible non-magical explanation for the event. The unstinting efforts of this office in persuading Muggles that all photographic evidence of the Loch Ness kelpie is fake have gone some way to salvaging a situation that at one time looked exceedingly dangerous.
Otherwise, memory charms, performed by a different sub-department, are used to wipe the memories of individuals. We hear of this happening in several books. Quoting again from Fantastic Beasts:
When the worst happens and a Muggle sees what he or she is not supposed to see, the Memory Charm is perhaps the most useful repair tool. The Memory Charm may be performed by the owner of the beast in question, but in severe cases of Muggle notice, a team of trained Obliviators may be sent in by the Ministry of Magic.
Both of these groups seem to be part of the same department, the "Department of Magical Accidents and Catastrophes, incorporating the Accidental Magic Reversal Squad, Obliviator Headquarters and Muggle-Worthy Excuse Committee" as revealed in OOTP.