RF receiver always having 1 on output
Though the datasheet doesn't say anything about it, the presence of a trimmer capacitor and lack of a crystal suggests that this receiver is a super-regenerative type. This uses positive feedback to increase the gain of a single transistor until it oscillates, then quenches the oscillation at a high audio frequency to produce extremely high (theoretically infinite) gain with very few parts. Because the gain is so high, if there is no signal it will simply output noise. 

This board also has an LM358 op amp, which converts the audio output of the super-regenerative detector into a 'digital' on/off signal. If the 'signal' is just noise then the digital output will simply toggle on and off randomly in time with the noise. An LED connected directly to the output will appear to be lit up continuously, when in reality it is flashing randomly at audio frequencies. 

When a strong enough RF signal is present the transmitted signal should appear on the receiver output. However unless the signal is very strong some noise may still be present. The device using the signal must reject this noise, by eg. only accepting pulses which have the correct shape and/or verifying that serial data matches its CRC. This can be done with suitable code in an MCU, or using a dedicated chip such as the Holtek HT12D