What does it mean to "push your values"?
If there isn't enough separation in the levels of the values used in rendering an image then the image can look washed out or the shapes poorly defined.

Someone saying, you need to "push the values", means they think you need to add more separation in the value levels you used to define your shapes and areas in the image as a way to help build greater depth, contrast and image intensity.

Here's a quick example



In the image on the left I limited myself to use values only from the middle range of the value scale (N4 to N7). The image looks hazy as though looking through a fog. I squeezed my values into too tight of a range for the scene.

For the image on the right I expanded the range and pulled from a broader overall range of the value scale. I pushed areas that were originally at N4 to now be at N1. At the other end where I had used N7 i pushed those areas to N10.

In both images N5 was used to render the table top. I used this value as my anchor point. Squeezing and pushing around it.

The result is the image on the right looks more intense and seems to be clearer than the image on the left.