Refer to figure. 
The aircraft altimeter has a rotating knob that allows the pilot choose the appropriate pressure setting in hectopascals (hPa). The altimeter then measures the current pressure at its location and compares it to the pressure set by the pilot. This pressure difference is then converted into altitude difference assuming ISA conditions. This is done automatically by the instrument. The pressure set by the pilot changes the pressure reference the altimeter uses to measure the altitude from. There are some common altimeter settings:

• QNH ⇒ the barometric pressure measured at an aerodrome which has then been mathematically adjusted down to mean sea level using the airfield's elevation. This adjustment assumes the ISA temperature for the station. QNH is used to obtain elevation when on the ground and altitude during flight.
• QFE ⇒ Atmospheric pressure at aerodrome elevation (or at runway threshold). QFE is used to get an indication of 0 ft when the aircraft is on the ground at the aerodrome, and the height above the aerodrome during flight.
• QNE ⇒ 1013.2 hectopascals (hPa) standard altimeter setting to obtain Flight Levels. It is used during flight to provide the same reference for all aircraft.

The SID requires the aircraft to turn at 500 ft AAL (Above Aerodrome Level). Therefore, QFE should be selected until the initiation of the turn. The pilot is cleared to climb to 7000 ft altitude, so the pilot needs to set QNH after the turn.

QNH → altitude above mean sea level
QNE → flight level above the 1013.25 hPa datum
QFE → height above aerodrome level

Let's examine the given options:

Request a flight level instead. → INCORRECT. The ATC clears the pilot to climb to an altitude of 7000 ft. Requesting a flight level to climb will not make any change to the need of the pilot to follow the restriction of 500 ft AAL (Above Aerodrome Elevetion).

Listen to the ATIS and set the QNH obtained from the ATIS. → INCORRECT. When QNH is set on the altimeter, altitude above mean sea level is provided. The pilot need to turn at 500 ft above aerodrome level, NOT above mean sea level. QNH may be used after initiating the turn at 500 ft AAL.

Select QFE. → CORRECT. When QFE is set, the altimeter indicates altitude above aerodrome elevation. With a QFE setting, the pilot may turn when the altimeter indicates 500 ft since this will be 500 ft AAL (Above Aerodrome Level). After the turn, the pilot may select QNH in order to climb to an altitude of 7000 ft.

Maintain standard pressure (1013.25 hPa) until passing 500 ft. → INCORRECT. When standard pressure (1013.25 hPa) is set, the altimeter indicates flight levels. The pilot needs to turn at 500 ft above aerodrome level, not at a flight level.