I have started out my first brew using a True Brew Kit All Malt Brown Ale

I have Brewed the wort and gone through the first day of fermentation.

For those not familiar with the True Brew products, The kit instructions say to leave the product in the fermintation vessel for the full 7 days and then bottle the product.

What is the advantage to transfering the product to a carboy after 3-4 days of fermentation.

Howdy,
Welcome to the wonderful world of brewing beer. A lot of brewers use a secondary fermenter so that the beer can continue to ferment away from much of the sediment that is produced by the yeast, hops, and grain. Moving the beer to a carboy will reduce the chances of off-tastes and will lead to a clearer beer. Often if you only use a single fermenter and bottle straight out of it, there will be some sediment on the bottom of each bottle. This sediment isn't harmful in any way but often brewers will try to reduce this by transferring their beer at some stage in the fermentation process. You certainly don't have to use a carboy to do a secondary fermentation. The beer will turn out just fine with just one. If you have a carboy, there's really no harm in transfering except for the slight danger of introducting bacteria into the beer that could be caused by using unsanitized equipment.

There are other brewers on this board who are far better brewers than I am and if you guys want to add something, that would be great. Cheffriz, I hope you keep us informed as to how the batch is coming along. Take care.

Welcome to the board! I often use just a single fermenting vessel, but I leave it there for 3 weeks. After 7 days your beer will probably be done fermenting, but will be very cloudy (which isn't a problem) and taste very yeasty (may be a problem depending on the style). 3 weeks seems to be enough time to let the yeast settle out and clean up any funky flavors.

If I do a higher gravity brew (above 6.5%), I'll move it to a carboy for a few days to few weeks at very cold temps to mellow and clear.

Hope this helps!

Every brewer has their own opinion on this. Here's my take:
First off, I don't think that the issue is time (i.e. Transfer to secondary after 3-4 days), so much as it is timing (i.e. Transfer to secondary after primary fermentation has finished and the trub has dropped out). The advantage here is, you get your beer off of the trub, which could potentially cause off flavors. Now your beer has a cleaner environment with which to clean up some of it's byproducts, like diacetyl, and drop out of your beer. Some people claim that leaving their beer on the trub can cause a soapy flavor. So, if you're getting this flavor, then racking to secondary after your trub falls out could be the answer.
I don't personally use a secondary unless I'm brewing a big beer that will be fermenting for a long time. Most of my beers spend 3-4 weeks in primary, after which I keg them. I have left beer in primary for up to 6 weeks with no bad flavors from trub or yeast autolysis, though I would recommend against trying this, if you can.
I'm pretty lazy when it comes to brewing beer, so I tend to go with whatever is easiest, so long as it doesn't negatively affect the final beer. Leaving my beer in 1 carboy until kegging time is the way to go for me. I personally haven't experienced any soapy flavors from trub. If I did, I probably still wouldn't secondary my beer, per se. 'Secondary' is short for secondary fermentation. If I had soapy flavors, I'd prefer to remove the trub from my beer before fermentation began. Therefore, I'd rack my beer into a carboy and crash cool it in an ice bath for 6 hours, and rack it to another carboy after the trub had all settled out. To me, this would be the best option of all, so long as you have a sufficiently large starter (or dry yeast) to prevent a long lag time. This way, the trub is out of your beer before you even begin fermenting, and any oxygen introduced by your racking would be a welcome addition at this point, rather than a thing to be avoided.

The great part about brewing is finding the system that works for you. Some folks love the process so much that they'll add any "extra" step they can. Some folks believe in KISS (keep it simple, stupid). And most of us land somewhere in the middle.

The best thing to do? Brew as much as you can and figure out what you like best!

I subscribe to the KISS method myself. If I'm doing a standard brew I let it sit in the primary the whole time then keg it. I used to do a secondary but I learned there is very little, if any, difference in the quality of the beer. You only increase your risk of contamination by adding this second step.