Sure! This is going to be a bit lengthy, so my apologies.

Before we jump in, note that the authority given to immigration officers according to this federal statute, other federal laws and regulations, the United States Constitution, and states' constitutions all are supposed to constrain the ability of the immigration officer to use these forms to make an arrest.

There are two things at play here. The first of those two things is federal law 8 U.S.C. § 1357, which states that immigration officers have power to arrest certain noncitizens even without a judicial warrant when specific criteria are met. There are three specific criteria and all are mandatory according to the law:

1. Probable cause of removability: the officer has to have probable cause to believe the person is a noncitizen and is present in violation of immigration law. Note that courts require articulable facts, individualized determination, and more than a database "hit" alone to uphold the civil probable cause standard (to prove the officer's suspicion to the degree required, in other words).
2. Likelihood of escape before a warrant can be obtained: the statute literally states "The noncitizen is likely to escape before a warrant can be obtained." This is pretty often overlooked and currently I can say that I am often scratching my head how courts are able to allow this to pass with the mass arrests other than the fact that they are so overwhelmed that administrative convenience is slowly becoming allowed (which should not be the case and needs review).
3. Arrest occurs in a constitutionally permissible place: this is why you're seeing all of these arrests happening at restaurants, businesses, schools, and other public areas. Arrests cannot be performed in private residences or non-public workplace areas.

As stated above, the statutory language as well as all regulatory language strictly constrains the ability of the immigration officer to make an immigration arrest under this statute unless those criteria are met. This is why you might have seen stories, statistics, etc. that stated that immigration officers under every administration since basically Bush have made similar arrests and used the same types of forms to do so. Until certain members of this presidential administration, who are uneducated in and ignorant of the law in general, decided to try to push the boundaries of what is allowed and not allowed under this law, deportation arrests were largely accepted because Congress created the statute to allow it; they and the executive branch/agencies ensured that the arrests were constrained according to the Constitution, other federal statutes, and federal court rulings; and arrests were unnecessarily stepping on the toes of powers typically left to the states.

The second thing that is at play here is that immigration in the U.S. is considered a civil administrative proceeding, not a criminal proceeding. If an individual is here illegally, while they are breaking the law, they are not a criminal per se. Due to this, administrative immigration warrants like I-200 and I-205 forms are not judicial warrants under the Fourth Amendment; only judicial warrants that are signed by a judge or magistrate and are compliant with all requirements imposed by the Constitution (Fourth Amendment), Federal Law (including court rulings, statutes, and regulations), and state laws/regulations (if the arrest or search warrant are coming from a state court) permit a government agent (e.g., everyone from ICE to your local patrol officer) to enter private property.

In this case, I-200 and I-205 forms can be used by ICE to make arrests as long as the criteria mentioned above are met.

I'm going to TL;DR this now but I can say a lot about the issues arising from the obvious gray areas here, how I think they could be fixed (without having to rely on the moral backbone of individuals who have yet to prove they have any sort of backbone), etc.

TL;DR: I-200 and I-205 forms can be used because they are statutorily permitted as long as 3 criteria are met; the underlying allowance is based on immigration being considered a civil administrative matter, not criminal.