Here is the context: Following that, a number of banks have cut their exposure to equities due to the volatile nature of stocks in the first half the year. Earlier this month, Goldman Sachs downgraded stocks to "underweight" as part of its 3-month asset allocation citing global equities to be at the upper end of their "fat and flat range".
Someone's (or, a bank's) "exposure to equities" refers to the amount of value which has a risk that fluctuates with the equities market (ie: the stock market).
In very broad terms, I think it might make sense to say that exposure to equities could mean, for example, owning many rental properties, if the rental market was "highly correlated" with the equities market. That is - if house prices go down when the equities market goes down, and if that relationship is very strong, then owning a house means you are exposed to the equities market.
However, in the sense it is used there, it seems to mean direct exposure to equities - ie: owning stocks and stock-based funds.
A "fat and flat range" refers to a view that stock indexes have been, and are likely to continue, fluctuating within a trading range, with fixed upper and lower bounds (contrasted with a trending market). The use of flat emphasizes this, because the trading range appears as horizontal channel on a chart of stock prices over time. The range is fat (wide) if the upper and lower bounds are far apart. This view would motivate caution (not investing as much in stocks) because stocks, starting at the upper end, would have a lot of room to fall within the range, and not much room to rise. It is similar to saying "resistance is nearby and support is far away".