more (adjective) than anything
In the example above, the speaker feels several emotions about getting her Ph.D. — pride, sadness that she's leaving the school, nervousness about her future, etc. But she's saying that the emotion she feels the most is relief.
Here's another example: if your baby is crying, you try changing her diaper, feeding her, holding her, etc. but she still continues to cry. You say:
I bet she's more sleepy than anything.
This means that sleepiness is the main thing that's making her cry.
You pronounce the sentence with the most stress on the adjective:
I feel more relieved than anything.
She's more sleepy than anything.
Here are a few more examples. Some of them use an adjective, but some use a noun after "more":We got you a little gift. It's nothing special. It's more a gesture than anything.
Don't worry about him. He's more hungry than anything. I'm sure his mood will improve once we get some food in his belly.
The political opposition to President Obama is more racism than anything else.