15:25a

But this is to fulfill what is written in their Law: This clause means “in this way what the Jewish scriptures said would happen has happened.” In this context it indicates that the people would hate Jesus and that would fulfill the Scriptures that said they would. See how you translated similar expressions in 12:38 and 13:18. For example:

This fulfills what is written in their Scriptures (New Living Translation (2004))
-or-
In this way what is written in their Scriptures has come true. (God’s Word)
-or-
That is why the Scriptures are true when they say (Contemporary English Version)
-or-
And so the words that were written in their scriptures were fulfilled which said

this: This pronoun refers to the fact that the people hated and rejected both Jesus and his Father (15:24d). Their hate fulfilled the Old Testament prophecy.

what is written in their Law: This phrase refers to the quotation from the Psalms that is quoted in 15:25b. It may be confusing to translate this phrase literally because the quotation contains several words. Here are other ways to translate this phrase:

the words written in their Law (New Jerusalem Bible)
-or-
what is written in their Law (New International Version)

in their Law: The pronoun their refers to the Jewish people. The phrase their Law here refers to their scriptures, the Old Testament in general. Jesus used the word their to emphasize that even their own law showed that they were wrong. It does not imply that the law does not have authority. Here are other ways to translate this phrase:

in their Scriptures (New Living Translation (2004))
-or-
what is written in the Holy Book of the law they follow (Otomi Back Translation)
-or-
even in their own scriptures it says

This quotation is taken from Psalm 35:19 and Psalm 69:4. Some translations will want to use a footnote to give this information. For example:

This quotation is from Psalm 35:19 and Psalm 69:4. Both verses contain these words.

15:25b

They hated Me without reason: This clause is the quotation from the Jewish scriptures referred to in 15:25a. In the Psalms, the pronoun They refers to David’s (the author’s) enemies. However, Jesus used it to refer to his own enemies, mainly the Jewish religious leaders. They had no good reason to hate Jesus. He had not acted out of hate for them but had helped them, healing their sick. Here are other ways to translate this clause:

They hated me for no reason at all. (Good News Translation)
-or-
They hated me without reason. (New Jerusalem Bible)

Me: In the Psalm, this pronoun referred to David, the author of the Psalm, but here Jesus applied it to himself.

without reason: This phrase means that Jesus did not do anything to deserve their hate. For example:

They hated me, but they had no reason to hate me
-or-
They hated me, even though I did nothing to cause them to hate me.

General Comment on 15:25a–b

In some languages it is more natural to combine 15:25a and 15:25b. For example:

25 Their law says, “They hated me without cause.” They are fulfilling that prophecy.

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