With a 220V primary and 6V secondary your transformer has a step-down ratio of 220/6 = 36.7. The primary resistance will be reflected in the secondary according to the square of the turns ratio. 1300Ω/36.72 = 0.97Ω. This appears in series with the secondary resistance, so the transformer's effective secondary resistance is 1.97Ω.
You got 3.75V With a 20Ω load, so the current draw was 3.75/20 = 188mA. Ignoring inductive effects, at this current the transformer should drop 0.188A*1.97Ω = 0.37V, which is only a 6% drop. Small transformers are typically rated for a maximum voltage drop of 10-20%.
However this simple calculation only applies to a purely resistive load. With a rectifier and filter capacitor the current will be drawn in short pulses during the peaks of the AC waveform, so voltage drop in the transformer will be higher. Your bridge rectifier will also drop 1.4-2V, and the capacitor voltage will droop between peaks.
Here's an LTspice simulation of your transformer/rectifier/capacitor, with a load drawing ~180mA.
The purple trace is the AC voltage output of the transformer. The green trace is the raw DC voltage on the capacitor. Blue is current through one diode, and red is load current.
Diode current peaks at 0.66A, which causes a loss of around 1V in the transformer and 1.9V in the rectifier bridge. Finally there is about 0.35V of ripple voltage across the capacitor, so the minimum raw DC output voltage is a miserable 5.0V. This leaves nothing for the regulator to work with. The 7805 needs about 1.5V of 'headroom' to overcome its dropout voltage.
To fix this you have a couple of options. The obvious solution is to get a transformer with higher secondary voltage, eg. 8V with bridge rectifier or 15V center tapped (7.5V-0-7.5V) with two rectifier diodes.
Alternatively you could keep the transformer and try to reduce losses in the rectifier and regulator. Schottky diodes drop about 0.4V (raising the raw DC output to 6V) and an LDO regulator could drop less than 0.2V, giving you about 0.8V of headroom. That's not a lot, but might be enough if your load is not bothered by occasional mains dropouts.