Websites want to connect with their visitors, which is why many of them lead off their homepages with an overt welcome. Type "Welcome to our site" into Google (keeping the quotation marks) and some 4 million results pop up. Meanwhile, many other companies include their company names in that phrase—for example, "Welcome to West End IT Solutions"—which puts the tally at many millions more. Ultimately, saying "Welcome to our site" on the homepage isn't all that welcoming, because it doesn't connect with visitors.
For one, no one wants to be welcomed by a computer. It's insincere. And second, the phrase is boring. When someone visits a site, they quickly want to know what the site is about. They want to know how the site can help them. Does it have information they can use? Does if have products they want to buy? The task of reading through bland welcoming text is an obstacle to them finding the answers they want. It's akin to the tedious talk of a salesperson trying to butter up a prospective customer. The customer may kind of be interested, and ultimately he or she just wants the conversation to get to the point.
The phrase is also a wasted opportunity. Visitors will quickly bounce out of a site if they determine the site isn't for them, so the copy on the home page should work to convince them that the site is indeed for them. The way to do that is to get to the point from the onset.
And if the site doesn't interest them, it will because they determined that to be the case, not because they didn't want to spend the time trying to figure out what the site is about.
To sum up, the best way to welcome visitors to a homepage is not to say "welcome" but to start giving them the information they are looking for.