Short answer: Yes, commercial table salt contains anti-caking agents; No, these do not prevent salt from dissolving in your body. Anti-caking agents are added to prevent the salt crystals from sticking together due to air humidity. They achieve this by reacting with water before the salt can. Without them you would have a hard time getting finely ground salt out of a salt shaker.
There are different agents in use. Sodium aluminosilicate seems to be the most common agent in the USA, while in Europe sodium and potassium ferrocyanide are used. They should be mentioned in the list of ingredients of any commercial salt you buy. None of these prevent salt from dissolving in liquid water, as you can easily demonstrate by adding a spoon of salt to a glas of water and stirring. The solubility of sodium chloride in water is about 360 g / liter at room temperature. I could not find any reputable sources claiming that this is changed by the addition of anti-caking agents. Sodium aluminosilicate is probably insoluble, depending on its exact composition (which can vary quite a bit), but that will only result in a suspension of the insoluble particles in your liquid.
Edit: The question whether these substances have a "biological effect" is different from the headline question. Since everything you interact with will have some kind of effect, it makes more sense to limit the answer to the question of toxicity. All anti-caking agents have to be and have been approved by the appropriate agencies (FDA in the US, EFSA in Europe).
Ferrocyanide is a highly stable complex that does not react in the body and is excreted via feces and urine. Its use as an anti-caking agent is limited to table salt in the EU. 1 At very high concentrations it can lead to kidney damage due to crystal formation in the kidneys. It has not been observed to have carcinogenic or teratogenic properties. 2 The LD50 of potassium ferrocyanide (oral, rat) is given as 3600 mg/kg, which is higher than that of pure sodium chloride (table salt) which is 3000 mg/kg. Since the highest allowed concentration of ferrocyanides in table salt is 20 mg /kg, it can be considered impossible to ingest a toxic dose of this agent. Criticisms against these anti-caking agents are usually based on possible environmental effects, but as such are not part of the question.
Sodium aluminosilicate is actually not a clearly defined compound, but includes several amorphous materials, zeolites and naturally occurring minerals. 3 It is generally insoluble in water and is excreted from the body via urine and feces. The peer reviewed literature shows no negative impact on health or environment. 4 (Toxnet entry, contains links to the peer reviewed literature) The LD50 data for this material is a bit inconsistent, giving values from >5000 to > 27000 mg/g (oral, rat). The highest danger comes from inhalation of the dust, but even here toxicity starts at 140 mg/m3 over 4 hours of exposure. 5
Since the other claims made in the quote were not part of the question, I will just provide a link to a blogpost where most of them are discussed.