Mara and Serengeti, where they are

In the Masai Mara National Park, located in southwest Kenya, the climate is pleasantly warm, with cool nights, all year round. The area is located just south of the Equator, but at an altitude between 1,500 and 1,900 metres (5,000 and 6,200 feet). Temperatures are slightly higher from October to March, while they are slightly cooler from June to August. At night it can be a bit cold, and the temperature can drop below 10 °C (50 °F), especially from June to August.
Here are the average temperatures.
Masai Mara - Average temperatures
Min (°C)121212131312111112121212
Max (°C)282827262525252527282727
Min (°F)545454555554525254545454
Max (°F)828281797777777781828181

Rainfall amounts to about 1,400 millimetres (55 inches) per year. The wettest month is April, the driest is July. In the two rainy seasons (see below), especially in April, the rains are abundant and can cause flooding.
To the south, across the border with Tanzania, the park continues in Serengeti; the whole area is famous for the spectacular migration of animals, savanna herbivores such as wildebeest and zebras, which move between the two parks, along a circular path throughout the course of the year, in search of pastures made green by the rains, and are preyed upon by lions, hyenas, leopards and cheetahs, while crocodiles and hippos wait for them in rivers. In the area we find the Mara River with its tributaries, and also some branches that are lost in the swamps.

Wildebeest migration in Masai Mara

In both parks there are two rainy seasons, one that lasts about six weeks and occurs from March to May (called "the long rains"), and the other that lasts about two weeks and occurs from October to December (called "the short rains"); however, between the two adjacent parks, the Serengeti, being located more to the south, remains quite rainy also in January and February, and becomes dry from June to September, while the Mara is quite rainy even from June to September, hence in this period the animals move from Serengeti to Mara, arriving usually in July, and then go back in October, when the season of the short rains begins in Serengeti. So in Masai Mara there is no real dry season, though it rains less from June to September, and July is the driest month.
Here is the average precipitation.
Masai Mara - Average precipitation

The animals do not move all together, but in herds of tens of thousands of individuals, and their movement can vary from year to year, depending on rainfall, and therefore of pasture availability; however, in Masai Mara, as we said, the animals are usually visible from July to October, with a higher probability in August and September.
Being the park near the Equator, the sun passes the zenith twice a year, in late March and late September, and remains very high above the horizon around noon also in the other months: therefore it requires appropriate protection, also because of the high altitude.

Lions in Masai Mara

The amount of sunshine in Masai Mara is good all year round, since the sky is often clear, although there may be some afternoon cloudiness, which can give rise to showers and thunderstorms, obviously more frequent in the two rainy seasons.
Masai Mara - Sunshine

The Masai Mara can be reached by flight, or by a car trip of 5-6 hours from Nairobi.

When to go

The best time to visit the Masai Mara goes from June to September, being the coolest, and (albeit slightly) the driest of the year. However, showers in the afternoon may still occur, and there can be a bit of cool (or even cold) at night, while during the day the weather is pleasantly warm.

Alternatively you can go in January and February, the other relatively dry season, when it's warmer, and as we said some showers are still possible.

As we have seen, the period in which the herds of wildebeest and zebras reach the park goes from July to October, with a preference for August and September. October is a bit hotter than the previous months, but on the other hand already in September there is a certain increase in temperature, due to the passage of the sun at its zenith; so it can be a little hot during the day, and less cool at night. For the rest, the rainfall amount is similar to that of the previous months, until the short rains season begins, which depending on the year can happen in October or November.
In January and February, you can still watch the local fauna, such as big cats.