Some details are given in Strategic Operations, which discusses the facings of the large craft from which the fighters are ejected. (They come out sideways to avoid the large craft from ramming the fighter from behind [drawbacks of front ejection] or incinerating the fighter [drawbacks of rear ejection]).
Specific details of how fighter/small craft bays are left, er, unspecific. This gives writers leeway to describe different launch mechanisms for different ships with different factions. A Federated Suns civilian JumpShip with a pair of small craft is unlikely to have the same launch/recovery mechanisms as a battle-tested Vengeance-class fighter carrier DropShip. Perfectly acceptable mechanisms include: catapults, the fighter's own reaction control system (RCS), or deck crew can push it out the side.
Some generalities, though:
1) BattleTech fighters will not be launched out the nose of a large vessel. While a nose launch would be safe if the carrier is coasting and the fighter is ejected by catapult or RCS, other situations result in bad things. If the carrier is running its engines (likely in a combat situation), it will be easily accelerating at 1G or more - i.e., adding speed faster than an F15 or F22 with its engines firewalled, possibly faster than the shuttle when its main tank is nearly empty. A fighter might launch with its own engines running, but BT fusion engines are such that a fighter could evaporate (or at least horribly maul) the carrier if it did so. On the other hand, a catapult launch at 120mph+ with the fighter's engines off only buys the fighter seconds before an accelerating carrier rams the fighter from behind. 2.5G = 0 to 120mph in 2.4 seconds.
2) BattleTech fighters might be dropped out the stern of the carrier, but they'll quickly need to dodge to avoid the kilotons-per-second drive plume if the carrier is running its engine. (If the engine is off, see point 1).
3) This leaves a sideways launch with some "oomph," such as from a catapult or fighter RCS. However, note that BT spacecraft are not very deep - you won't get a long, Battlestar Galactica-type tunnel launch (see DropShip cutaways in DropShips & JumpShips). You'll get a quick heave-hoe from the short fighter bays to get your fighter out the ship and with a little sideways velocity to keep you clear of the carrier's drive plume. Ideally, the fighters will quickly pivot to get their carrier-wrecking asses aimed away from the carrier.
4) Civilian carriers are unlikely to launch craft frequently under thrust, so they don't need catapult assists. They can puff their way out with RCS.