Despite the understandable misconception of many new to Airsoft, a BB being fired does not (or at least should not anyway) touch the inside of the inner barrel. The air / gas pressure behind a pellet pushes the back of the pellet hard but also slips around the pellet escaping forward through the gap between the inner diameter of the inner barrel and the outer diameter of the BB pellet. This tight flow of air around the pellet prevents it from touching the sides thus a pellet being fired is actually riding down on a cushion of air for minimum friction and maximum acceleration.
A tightbore barrel reduces the inner diameter of the barrel in an attempt to reduce the space between barrel and pellet resulting in an even tighter air flow. This tighter seal against the pressure means that less air / gas can escape so the pressure behind and around the pellet is greater which leads to more efficient transfer of pressure to acceleration as well as exerting a greater trapping force between pellet and barrel for a more stable, straighter acceleration pathway.
6.03mm diameter inner barrel are already tightbore enough for most uses but the 6.01mm model pushes the envelope. Because the size is so incredibly close to the BBs diameter you have to make sure to use the right kind of ammunition, premium brands only. Some brands have size tolerances that vary too much or have average sizes too large such that they would simply jam in the barrel or at least be large enough to scrape the sides as they travel down the barrel which greatly increases wear and tear as well as significantly reducing power and consistency.
So that all said, apologies for the mini essay but some newer or less knowledgeable players need to have that explained. If you want to step up your inner barrel to a tightbore, PDI is a truly the premium brand for top tier modification. In particular, the PDI 6.01mm is probably the best series of premium barrels around. Just make sure you use the right kind of ammo otherwise you will at least negate the advantage of this barrel if not even under perform and possibly damage the performance and / or barrel.
Furthermore, do be sure to pick up the right kind of barrel; PDI makes both 6.03 and 6.01 inner barrels in various different lengths for various different models of weapon so do be sure to pick the right type and length for you.