It's about as easy to explain on paper (or on a website) how to climb a technical trail as it is to, say, explain the offside rule in football (soccer) to someone who insists on calling it soccer. Each climb will be unique and have its own challenges. The following should just give you some broad pointers.
It’s Not About Power
Hammering on the pedals won't always work. If the climb is technical then it probably means there are all sorts of rocks, roots, steps and low-traction surfaces on it. Being a brute won't help you too much. Instead focus on finesse, timing, poise and balance.
You probably won't be moving very fast if you are on a technical climb so having good balance is a vital skill. You'll be dealing with the trail underneath you and you'll need a considerable capacity to stay upright at low speeds. A good practice is to learn to hold a good track stand or rolling as slow as you can without putting a foot down. Another practice is to learn to hop up and over curbs, benches, rocks or any stationary obstacle. Just play around and it will tune your balance instinct before you know it. Remember, dick about at every opportunity.
Ratcheting refers to executing incomplete pedal strokes, short quarter or half strokes that propel you forward somewhat but also keep your pedals from striking the ground on tricky terrain. Learn to see the power in pushing through your bike to maintain balance or momentum and pushing powerful pedal bursts rather than mashing the pedals.
Move Forward on Seat, Get Shoulders Down
See that big, fat seat? Well, use all of it. On steep technical climbs it might not be possible to stand up without losing traction on the rear wheel so you'll need to keep perched on your saddle. Sliding forward to the nose of the saddle on steeper climbs allows you to maintain weight on the rear tire and, at the same time, keep some weight on the front wheel so it doesn't lift, become weightless, wander about or flip you backwards.
As you slide forward on the seat also consider lowering your shoulders to the bars a little way to keep your center of gravity low, push more weight into the front tire and preload your arms, shoulders and body if you need to lift the front wheel to get over objects on the climb.
Slow Down or Stop Before
It's a good idea to sometimes take a little breather before a challenging climb. It will give you a chance to get your breath back, look at the trail ahead, assess the climb and find the line you want to ride. Remember to play to your own strengths and ride what you think is a good line. Watch what line people ride and see how successful it is but also leave room to consider options that others don't try because you might be an entirely different kind of rider than your buddies.
Of course, you might not always be able to do this, you might be racing! In which case, you better get better quickly or learn to dismount and run fast. Go!
Gear Before Obstacle
Part of pausing before the climb will be to decide what is a good gear to attempt it in. Even though modern drivetrains allow us to shift gears under load like never before, you don't want to be testing the survivability of your chain and shifting prowess of the derailleur if you find yourself in the totally wrong gear and try to throw a few gears mid-climb. Figure out what gear you can push in different circumstances and remember to shift early and get into a good gear.