The simplest answer would probably be that they only reject the specific Oral Torah of Rabbinic Judaism; they don't reject interpretation altogether. This is essentially what R. David Ibn Zimra writes in his responsum about Karaite Torah scrolls:
Shu"t Radvaz 2:774
ואני שאלתי את פי הגדול שבהם והוא בקי בכ"ד ספרים על עניינים הרבה שאינם מפורשים בתורה כעין קדושין וגטין ושחיטה שהם נוהגין בהם כל אלו מנין לכם ואמר לי כך למדו רבותינו ואמרתי לו א"כ למה אין אתם מאמינים בכל דברי רז"ל אמר לי כל דבר שאינו מנגד פשטי הכתובים אנו מאמינים להם וכל דבר שהוא מנגד פשטי הכתובים אין אנחנו מאמינים אמרתי לו תפח רוחו אם כן תורתכם בחיריית תופסים מה שהוא נאות לכם ומניחים מה שאינו נאות נמצא כל אחד מכם מפרש במצות הכתוב מה שירצה ויתיר מה שירצה ויאסור מה שירצה שהרי אין לדתכם שום גדר וסייג ולא שום מדה מהמדות שהתורה נדרשת בהם כללות הדברים הכחישו הקבלה והרי הם כעורים באפלה לא ידעו במה יכשלו ועליהם אמר הכתוב ורשעים בחשך ידמו
And I asked the mouth of the greatest of them — and he was an expert in the 24 books — about many matters that are not explicit in the Torah, such as marriage, divorce, and slaughter, all of which they [Karaites] practice, "whence do you know?" And he said to me, "such have our teachers taught us". And I said to him, "if so, why do you not believe in all the words of the Sages?" He said to me, "anything that is not against the plain meaning of the verses, we believe them; anything that is against the plain meaning of the verses, we do not believe them." I said to him, "blast his soul. If so, your Torah is free. You can grasp whatever is pleasant for you and leave aside whatever is not pleasant. It comes out that each of you interpret the commandments of the Torah however he wants, and permits whatever he wants, and forbids whatever he wants, for behold your religion has no wall or fence, and none of the methodologies by which the Torah is expounded." The general principle is: they deny the Tradition, and they are like blind people in the dark; they don't know in what they are stumbling, and about them Scripture states, "and the wicked in darkness shall be silent."
This general idea is also noted by Karaites themselves:
As it is Written: A Brief Case for Karaism p. 29
Do Karaites believe that everything contained in the Talmud is wrong?
No: To argue that every interpretation contained in the Talmud is wrong would be the same as arguing that a view is wrong simply because it is possessed by Karaites. This is not a fair means of debate, nor is it conducive to finding the truth behind some of the more pertinent issues in the Tanakh. When evaluating an interpretation, one must evaluate the interpretation rather than the person who holds the interpretation. In this case, it is fair to say that we should look at the message and not the messenger. On the other hand, to read the Talmud with the preconceived notion that it is the final word on the issues at hand is unacceptable — especially since the Talmud often reaches conclusions devoid of biblical support or even directly in contradiction to the Torah.
So in short, they can accept anything in the Oral Torah as long as it does not contravene their methodology of interpretation. This methodology is explained more earlier, on p. 14:
The fundamental tenet of Karaism is thus: "Search in the Scriptures well and do not rely on anyone's opinion." Since in the end, we are accountable for our own decisions and actions, it is important that we each loo into the religious issues that concern our daily lives. While Karaites are encouraged to seek the opinion of others, it is imperative that we not accept those opinions without verifying that they make sense in the context of the Biblical narrative.
For a specific example of how the Karaite interpretation of a commandment differs from the Rabbinic interpretation, you can see my answer about Karaite slaughter. For an example where the Karaite interpretation came to the same conclusion as a Rabbinic interpretation, you can see my answer about tefillin.
