I used APKTool to find the following details:

1. It’s not a framework or a shared library,
2. It is not supported,
3. It seems to use some custom permissions (INTENT_FILTER_VERIFICATION_AGENT, BIND_INTENT_FILTER_VERIFIER, INTENT_FILTER_NEEDS_VERIFICATION)
4. It has a transmission receiver, so presumably this is happening at an operating system level,
5. It is a service.

I tried to read the inverted code, but I have no experience with Smali and, therefore, I did not make significant progress. Then, I used AndroGuard to make a Call Chart:

androcg.py --output intent-filter.gml --no-isolated Intent Filter Verification Service_com.android.statementservice.apk

… then I turned this into an image:

import networkx as nx
import matplotlib.pyplot as plt
ifv = nx.read_gml('intent-filter.gml')
# filter out some system stuff, I think
ifv = ifv.subgraph((a for a in ifv.nodes() if not 'Landroid' in a))
# filter out some java utility type stuff, I think
ifv = ifv.subgraph((a for a in ifv.nodes() if not 'Ljava/lang' in a))
isolates = nx.isolates(ifv)
nonisolates = list(set(ifv.nodes()) - set(isolates))
ifv = ifv.subgraph(nonisolates)
relabeler = {key: key.split()(0).split('/')(-1) for key in ifv.nodes()}
ifvn = nx.relabel_nodes(ifv, relabeler)
ifvn.remove_edges_from(nx.selfloop_edges(ifvn))
nx.draw(ifvn, with_labels=True)
fig = plt.gcf()
fig.set_size_inches(18, 10)
fig.savefig('intent-filter.jpg',dpi=200)

I understand this:

My suspicion for the name of the service is that it effectively takes a Intent and determine if that or not Intent satisfies the (grammar? of …) Intentional filter of each exportable component declared publicly in each Manifest.xml for all applications on the device. This call chart seems to support my hypothesis. In particular, URL it is a stalemate with a ton of protruding edges, the nodes in the middle ring of things are seen as verification or filtration of some kind (analyzing an extended nano message, looking at a network response, extracting data from the Intent, obtain a web port, verify if a protocol is valid or invalid …) and, finally, the most external elements that only have incoming borders seem to be executing the Intent passing it to a destination (for example, HttpURLConnection ->, NetworkResponse, AndroidAppAsset, …), or discarding it as invalid (InvalidProtocolBufferNanoException, ResistanceFollowRedirects, OutOfSpaceException, …)

I'm just an amateur detective and I can't say that I really know what all this means, but these are my 10 cents. I believe that the Service forms the connection between a summoned Intent and all the IntentFilters in applications on the device. I think it decides what goes to which application, if something has invalid data, as in a null scenario, and if something should go to the web. I think it also stops some kind of infinite redirection attack. I suspect that this is an effort by the Android development team to defend against attacks between component communication (ICC) and application link vulnerabilities / URIs. I hope this helps!