Why do laptops need larger transformers than mobile phones?

All modern AC adapters or DC supplies are switched mode circuits/systems. For safety, the AC line maybe isolated with a transformer. It is a high frequency transformer, thus much smaller in physical size.

AC is 50/60Hz (cycles per second). Switching regulators are 50kHz to Mega-Hz. As such, the isolation transformer is much smaller. This is the reason of the change from massive transformer to a much smaller high-kilo-Hz transformer.

The material savings (copper winding, iron core), and the efficiency by electronic switching, effecting much lower cost, much more energy efficient, and, smaller size.

Same as old transformer design here: The 'output' side (2ndary) of the transformer is rectified to raw DC Voltage. For smallest size, the transformer coil ratio might be 1:1 (output at 110VAC,USA). High Voltage! Or whatever ratio for best overall design. The difference: The raw DC is the DC power supply for a switching circuit only, not to the output. The switched circuit output is the final DC supply.

Switched circuit simplified: When the switch is on, the raw DC charges the coil. When off, raw DC is disconnected from the coil. Now, by nature of coil, the coil forces the energy out of itself (try relieving itself!). The switches at its terminals 'happen' to be on and connected to a capacitor. The coil dumps its energy to the capacitor. This capacitor is the output DC smoothing capacitor, doubling as a 2ndary energy storage.

The load at the output, mean while, continues to deplete the capacitor energy. The coil recharges the capacitor from time to time. The raw DC replenishes the coil energy, from time to time.

In non-isolated case, no transformer, and the AC 110V (USA) is directly rectified (dangerous high Voltage!) to form the raw DC (about 120-150Vdc). 

The rest of the electronics regulate the output Voltage. When the capacitor reaches the desired Voltage the coil is switched off from the capacitor, preventing from charging to higher and higher Voltage. At the same time, the coil is reconnected to raw DC to recharge. When output is depleted too low, the coil is reconnected back to the capacitor to recharge it.

The switching frequency is chosen for optimum results, considered among physical size, efficiency, and cost.

In summary: Rectify; high DC Voltage; charge the coil; dump the coil energy to output capacitor; repeat.

By nature, switching circuit is NOT isolated (DC to DC switching). At least one wire is common, a direct connection from input to output.

If isolation is not needed (say, inside a closed package, such as a light bulb), maybe no transformer. Isolation is for safety, so a transformer is added. The lower the frequency, the less efficient in electric-magnetic conversion. Surely, at too high frequency, conversion efficiency begins to tail off.) Coil summary: One optional isolation transformer. At least one coil to store energy as a way to transfer energy from input to output.

Extra for the inquiring mind: Skip the coil! All you need is a switch to charge the output capacitor (switched capacitor mode!), directly from the raw DC! When reaching the desired output Voltage, switch off. Done! Save a coil component! You'd say: Cannot Voltage drive a cap? OK, add a current limiting resistor. Resistor is still much cheaper than a coil. Why need a coil? More ... Why not raw rectify the AC 110V, then the raw DC supply for a high frequency generator to drive a high frequency transformer? Instead of 60Hz, you now have 50kHz AC system! Same small transformer. 

Next, transformer steps down the AC Voltage. Rectify, Voila! [Hint: efficiency, and output power].

[Efficiency: Energy on capacitor=(1/2)xCV^2; coil equivalent: (1/2)Li^2. As Voltage gets higher on the cap [or equivalent for coil], it is more efficient: V is squared. Square 5V=25. Square 100V=10,000! Dumping 5V to the capacitor/coil is only that much. Dumping 105V (110V-5Vout) on a coil, wow!]