How can a probability density be greater than one and integrate to one
Consider the uniform distribution on the interval from $0$ to $1/2$. The value of the density is $2$ on that interval, and $0$ elsewhere. The area under the graph is the area of a rectangle. The length of the base is $1/2$, and the height is $2$ $$ \int\text{density} = \text{area of rectangle} = \text{base} \cdot\text{height} = \frac 12\cdot 2 = 1. $$
More generally, if the density has a large value over a small region, then the probability is comparable to the value times the size of the region. (I say "comparable to" rather than "equal to" because the value my not be the same at all points in the region.) The probability within the region must not exceed $1$. A large number---much larger than $1$---multiplied by a small number (the size of the region) can be less than $1$ if the latter number is small enough.
Remember that the 'PD' in PDF stands for "probability density", not probability. Density means probability per unit value of the random variable. That can easily exceed $1$. What has to be true is that the integral of this density function taken with respect to this value must be exactly $1$.
If we know a PDF function (e.g. normal distribution), and want to know the "probability" of a given value, say $x=1$, what will people usually do? To find the probability that the output of a random event is within some range, you integrate the PDF over this range.
Also see Why mvnpdf give probablity larger than 1?
I would like to point out an extreme example. Consider a probability density
\begin{equation}
f(x) =
\begin{cases}
\frac{1}{2d},& -d \leq x\leq +d \\
0, & \text{elsewhere}
\end{cases}
\end{equation}
Now, $\forall ~ d\neq0$
\begin{equation}
\int_{-\infty}^{+\infty} f(x)dx = 1
\end{equation}
Start with a smaller $d$ and find the integral again. It will be the same. It gets really interesting when we take the limit when $d \rightarrow0$ . The integral is again 1, but how does $f(x)$ looks like?
\begin{equation}
\lim_{d \to 0} f(x) =
\begin{cases}
\infty, & x=0 \\
0, & \text{elsewhere}
\end{cases}
\end{equation}
This limit is termed as the Dirac delta function $\delta(x)$, which is not really a function but the limit of a function.
So the bottom line here
Not only the probability density can go greater than $1$, it can assume even bigger values (the biggest one is noted here) as long as the area under it is $1$.