From the same press release:
"Johannesburg is a high-altitude airport. Climactic conditions during the hot summer in the Southern Hemisphere favour the operation of the larger and more powerful B747-400, which Cathay Pacific plans to deploy with the commencement of the additional flight. During the cooler winter months the airline's Airbus A340-300 is a more cost-effective option."
What CX is saying here is that the A340-300 is a more economical aircraft (which it is) but that it will not cope with the hot and high conditions with a full load in the Summer Months. I imagine on hotter days the aircraft would have a PLTOM (performance-limited takeoff mass) lower than it's MTOM (maximum [structural] takeoff mass), and this may lead to load restrictions.
On the other hand, the 747-400 has a higher capacity and therefore may also be suited to the Summer season if more passenger demand exists during those months.
But in direct response to your question the weather, particularly the temperature and pressure altitude, has a significant effect on aircraft takeoff performance, leading quite commonly to takeoff mass restrictions (the PLTOMs I mentioned earlier).
Take the Airbus A320 out of Birmingham, UK (EGBB). On a nice cold winter day (0 degrees C) the aircraft can take off with TOGA thrust, packs off and Flaps 3 at a maximum mass of 71,500kg. Knock the temperature up to 32 degrees C (bit unrealistic for Birmingham but not some airports) and you have a new PLTOM of 69,800kg which will either reduce your range or your payload. With a much more unrealistic 40 degrees C (Phoenix, USA-style temperatures) the PLTOM reduces to 65,400kg.
If you are really not worried about range / payload then with Flaps 1 and Packs on @ 40 degrees C your PLTOM is down to 63,600kg.
I used to love the smell of Jet-A in the morning...