1. During the evaluation process, qualitative and quantitative judgments about the extent to which the material and methods satisfy the external and internal criteria must be made.
"...the extent to which..." here means "how well".
"Qualitative" relates to how good something is (quality). "Quantitative" relates to how much of it there is (quantity).
So it means "judgements will be made about how well the material and methods satisfy the internal and external criteria, in terms of quality and quantity".
For instance, if the scenario is "delivering fresh milk to the supermarket", it might work out as follows:
1. Material – milk.
2. Methods – delivery vans.
3. Internal criteria – the standards the farmer sets for his milk production.
4. External criteria – the standards the supermarket sets.
5. Qualitative judgements – how good the milk is.
6. Quantitative judgements – how much is produced.
Is that any clearer?