I had another part written up but I lost the connection. Here it is:
You have this function \(y=f(x)\). This notation is used to mean that \(y\) is some value that depends on the value of \(x\). The function \(f\) is the matching rule that assigns any given value of \(x\) (so long as it's in the domain) to a particular value of \(y\).
The \(\Delta\) symbol is traditionally used to denote the notion of change. (\(\Delta\) is the Greek uppercase letter \(D\); think "D" for "difference".) For instance, if \(y=f(x)\) denotes the temperature \(y\) as a function of time \(x\), then \(\Delta x\) can be thought of as the change in time, i.e. given two specified instants of time \(x_1\) and \(x_2\) (where \(x_2>x_1\)), you have \(\Delta x=x_2-x_1\). (The condition that \(x_2>x_1\) isn't necessary, but it's often useful to use this definition for \(\Delta x\), called the "forward difference".)
Since \(y\) is a function of \(x\), then it's reasonable to believe that any change that occurs in \(x\) will result in some change to occur in \(y\).
So we have some function \(y=f(x)\) that's sensitive to the value of the independent variable \(x\). If \(x\) changes, i.e. we add some amount \(\Delta x\) to \(x\), then we have some new value \(x+\Delta x\).
Now if we apply our function, we have \(f(x+\Delta x)\). We don't how this affects \(y\) exactly unless we have the specific information regarding \(f\) and the amount of change \(\Delta x\), but we can predict \(y\) to change appropriately by some amount \(\Delta y\). So, you end up with what you're given in that image:
\[y+\Delta y=f(x+\Delta x)\]