A tRNA molecule at body temperature travels at roughly 10 m/s. Assuming a point-sized tRNA and stationary ribosome of radius 125 * 10^-10 m, the ray casted by the moving tRNA will collide with the ribosome when their centers are within 125 * 10^-10 m of each other. The path of the tRNA sweeps a "collidable" circle of the radius of 125 * 10^-10 m, for a cross-sectional area of 5 * 10^-16 m^2. Multiplied by the tRNA velocity, the tRNA sweeps a volume of 5 * 10^-15 m^3 per second. Constrained inside an ordinary animal cell of volume 10^-15 m^3, the tRNA would have swept the entire volume of the cell five times over in a single second. Obviously the collision path would have significant self-overlap, but at this rate it's quite likely for the two to collide at least once any given second.
Now, consider that this analysis was only for a single ribosome/tRNA pair. A single ribosome will experience this collision rate multiplied by the total number of tRNA in the cell, on the order of thousands to millions. If a ribosome is bombarded by tens of thousands of tRNA in a single second, it's very likely one of those tRNA will (1) be charged with an amino acid, (2) be the correct tRNA for the current 3-nucleotide sequence, and (3) collide specifically with the binding site on the ribosome in the correct orientation. In actuality, a ribosome synthesizes a protein at a rate of ~10 amino acid residues per second.
Any given molecule in the cell will experience millions to billions of collisions per second. The fact that molecules move so fast relative to their size is what allows these reactions to happen on reasonable timescales.