To check the files, first import the Armory signing key (4922589A). Then download the files. For each file, take its sha256 hash and then check that the hash matches the one in the signed hash file. Then verify the signed hash file.
For example, in Ubuntu, to verify the .dev file for 0.94.1, do the following.
$ gpg --recv-keys --keyserver keyserver.ubuntu.com 4922589A gpg: requesting key 4922589A from hkp server keyserver.ubuntu.com gpg: key 4922589A: public key "goatpig (Offline signing key for Armory releases) <[email protected]>" imported gpg: 3 marginal(s) needed, 1 complete(s) needed, PGP trust model gpg: depth: 0 valid: 2 signed: 0 trust: 0-, 0q, 0n, 0m, 0f, 2u gpg: Total number processed: 1 gpg: imported: 1 (RSA: 1) $ sha256sum -c sha256sum.asc.txt armory_0.94.1_amd64.deb 2>&1 | grep OK armory_0.94.1_amd64.deb: OK $ gpg --verify sha256sum.asc.txt gpg: Signature made Sun 22 Nov 2015 01:34:46 AM EST using RSA key ID 4922589A gpg: Good signature from "goatpig (Offline signing key for Armory releases) <[email protected]>" gpg: WARNING: This key is not certified with a trusted signature! gpg: There is no indication that the signature belongs to the owner. Primary key fingerprint: 745D 707F BA53 968B DF63 AA8D 8C52 1176 4922 589A