To eke out in this context means that it was difficult and took a lot of effort. Maybe because she was very nervous and she found it difficult to speak. You can use the phrase in examples such as: The toothpaste tube was almost empty but I managed to eke enough out to brush my teeth. Or, it can mean to use something frugally or economically: If we eke the toothpaste out, it should last us until we go to the supermarket at the weekend.

'eke' means to barely be able to do something. You did it but it was done with much difficulty.